Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #50167 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,4)(3,7,8,6)(11,17)(15,21)(18,23)(22,24)(25,26,31,29)(27,35,36,32)(28,39,42,33)(30,43,44,34)(37,45,40,47)(38,46,50,49)(41,48,54,52)(51,56,53,55);(1,3,5,8)(2,6,4,7)(10,14)(15,21)(16,19)(22,24)(26,29)(27,36)(33,39)(34,43)(37,40)(38,50)(48,52)(51,53);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,31,36)(26,32,29,35)(28,40,42,37)(30,38,44,50)(33,47,39,45)(34,46,43,49)(41,53,54,51)(48,56,52,55);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,33)(27,37)(29,39)(30,41)(31,42)(32,45)(34,48)(35,47)(36,40)(38,51)(43,52)(44,54)(46,55)(49,56)(50,53);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,29,31,26)(27,35,36,32)(28,39,42,33)(30,43,44,34)(37,47,40,45)(38,49,50,46)(41,52,54,48)(51,56,53,55);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,44)(26,34,29,43)(27,38,36,50)(28,41,42,54)(32,46,35,49)(33,48,39,52)(37,51,40,53)(45,55,47,56)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818197);